For purposes of discussion and simplicity, the effects of cascade defect clustering and recombination are ignored, and we consider only single mobile vacan­cies and SIA defects in the simplest form of RT to illustrate the CBM. At steady state, isolated vacancies and SIA are created in equal numbers and annihilate at sinks at the same rate. Dislocation-SIA interactions due to the long-range strain field result in an excess flow of SIA to the ‘biased’ dislocation sinks and, thus, leave a corresponding excess flow of vacancies to other neutral (or less biased) sinks, (DvXv — DiXi). Here, D is the defect diffusion coefficient and X the corresponding atomic fraction. Assuming that the defect sinks are restricted to bubbles (b), voids (v), and dislocations (d), the DX terms are controlled by the corresponding sink strengths (Z): Zb («4ягЛ), Zv («4prvNv) for both vacancies and SIA; Zd («p) for vacancies and Zdi («p [1 + B]) for SIA. Here, r and N are the size and number densities of bubbles and voids, p is the dislocation density, and B is a bias factor. At steady state,

DvXv — DiXi = [BGdpaZdi]/{(Zb + Zv + Zd)

(Zb + Zv + Zd[1 + B])} + DvXve [2]

Here, DvXve represents thermal vacancies that exist in the absence of irradiation and p («1/3) is the ratio of net vacancy to dpa production. In the absence of vacancy emission, the excess flow of vacancies results in an increase in the cavity radius (r) at a rate given by

dr/dt+ = (DxXv — DiXi)/r [3]

However, cavities also emit vacancies, resulting in shrinkage at a rate given by the capillary approxi­mation as

dr/dt— = — DvXve exp[(2g/r — p)Q/kT]/r [4]

The Xveexp[(2g/r—p)O/kT] term is the concentra­tion of vacancies in local equilibrium at the cavity surface, and O is the atomic volume. Thus, the net cavity growth rate is

dr/dt ={DvXv — DiXi — DvXve

exp[(2g/rc — p)O/kT ]}/r [5]

Growth stability and instability conditions occur at the dr/dt = 0 roots of eqn [5], when

DvXv — DiXi — DvXve exp[(2g/r — p)O/kT] = 0 [6a]

Note that DvXve is approximately the self-diffusion coefficient, Dsd. The He pressure is given by

p = 3kmkT/4%r3 [6b]

Here, к is the real gas compressibility factor. Equa­tion [6a] can be expressed in terms of the effective vacancy supersaturation,

L =(DVXV — DiXi)/Dsd [6c]

The bubble and critical radius occur at

L — exp[(2g/r—p)O/kT ]=0 [6d]

In the absence of irradiation (or sink bias), L = 1 and all cavities are bubbles in thermal equilibrium, at p = 2g / rb. Assuming an ideal gas, к = 1, eqn [6d] can be written as

2g/r — (3 mkT )/(4nr3) — kT ln(L)/O = 0 [7 a]

Note that kT ln(L)/O is equivalent to a chemical hydrostatic tensile stress acting on the cavity. Rear­ranging eqn [7a] leads to a cubic equation with the form,

rc3 + C1 r2 + C2 = 0	[7b]
c1 = — [2gO]/[kT ln(L)]	[7c]
c2 = [3mO]/[4p ln(L)]	[7d]

As shown in Figure 2(d) and 2(e), eqn [7b] has up to two positive real roots. The smaller root is the radius of a stable (nongrowing) bubble containing m He atoms, rb, and the larger root, r*, is the corresponding critical radius of a (m*,n*) cavity that transforms to a growing void. Voids can, and do, also form by classical heterogeneous nucleation

rc (nm) Figure 11 The CBM predictions of radial growth rate of cavities as a function of their He content, m, normalized by the critical He content for conversion of bubbles to growing voids, m*. The effective supersaturation is (L = 4.57), temperature is (T = 500 °C), and surface energy is (g = 1.6 J m—2). The two roots in the case of m < m* are for bubbles and voids, respectively. Cavities can transition from bubbles to voids by classical nucleation or reach a m* by He additions. The effect of He on the growth of voids is minimal at sizes larger than about 2.5 nm in this case.

on bubbles between rb and r*.109,132,141 However, as shown in Figure 2(d) and 2(e), as m increases, rb increases and rv decreases, until rb = rv = r* at the critical m*. An example of the dr/dt curves assuming ideal gas behavior taken from Stoller133 is shown in Figure 11 for parameters typical of an irradiated AuSS at 500 °C with L = 4.57. The corresponding r* and m* are 1.50 nm and 931, respectively.

The critical bubble parameters can be evaluated for a realistic He equation of state using master correction curves, 01(ln L) for m* and 02(ln L) for r*, based on high-order polynomial fits to numerical solutions for the roots of eqn [7b].143 A simpler ana­lytical method to account for real gas behavior based on a Van der Waals equation of state can also be used.1 1 The results of the two models are very simi­lar.143 Voids often form on critical bubbles located at precipitate interfaces at a smaller m* than in the matrix.142 This is a result of the surface-interface tension balances that determine the wetting angle be­tween the bubble and precipitate interface (see Figure 20(b)). Formation of voids on precipitates can be accounted for by a factor Fv < 4я/3, reflecting the smaller volume of a precipitate-associated critical

bubble at r*, compared with a spherical bubble in the matrix, with Fv = 4я/3. Note that the critical matrix and precipitate-associated bubble have the same r*. The m* and r* are given by

m* = [32Fv01g3O2]/[27(kT)3(ln(L)2)] [8a]

r* = [402gO]/[3kT ln(L)] [8b]

Figure 12 shows m*, r* as a function of temperature for typical parameters for SA AuSS steels taken from Stoller.13 More generally, L can simply be related to Dsd, V, Gdpa, B, and the sink’s various strengths. Assuming Zv « 0 during the incubation period,

L «{[vGdpaBZd]/[(Zb + Zd)

[9]

(Zd(1 + B) +Zb)]/Dsd} +1

Figure 13 shows the corresponding m* and r* as a function of the concentration of 1 nm bubbles, Nb, at 500 and 600 °C again using the AuSS parameters given in Stoller.133 Clearly, high Nb can lead to large critical bubble sizes requiring high He contents for void formation.

Thus, to a good approximation, the primary mechanism for void formation in neutron irradiations is the gradual and stable, gas-driven growth of bub­bles by the addition of He up to near the critical m*.

Although nucleation is rapid on bubbles with m close to m*, modeling void formation in terms of evaluating the conditions leading to the direct conversion of bubbles to voids is a good approximation.1 The corresponding incubation dpa (dpa*) needed for Nb bubbles to reach m* is given by

dpa* = [m* Nb]/[He/dpa] [10]

Figure 14 shows dpa* for He/dpa = 10appmdpa-1 and the same AuSS parameters used in Figure 13. Clearly high Nb increases the dpa*, both by increas­ing the neutral sink strength, thus decreasing A, and partitioning He to more numerous bubble sites. Indeed, in the bubble-dominated limit, Zb >> Zd and Zv the dpa* scales with N^l

The CBM also predicts bimodal cavity size dis­tributions, composed of growing voids and stable bubbles. Once voids have formed, they are sinks for both He and defects, and thus slow and eventually stop the growth of the bubbles to the critical size and further void formation. Figure 15 shows a bimodal cavity versus size distribution histogram plot for a Ni-He dual ion irradiation of a pure stainless steel,114 and many other examples can be found in
the literature111,112,114,133,153 Figure 16(a) shows low He favors the formation of large voids in a CW stainless steel irradiated in experimental breeder reactor-II (EBR-II) to 40 dpa at 500 °C and 43 appm He, resulting in «12% swelling, while Figure 16(b) shows that the same alloy irradiated in HFIR at 515— 540 °C to 61 dpa and 3660 appm He has a much higher density of smaller cavities, resulting in only

2% swelling.16

Thus, while He is generally necessary for void formation, very high bubble densities can actually suppress swelling for the same irradiation conditions as also shown previously in Figures 9 and 10. This can lead to a nonmonotonic dependence of swelling on the He/dpa ratio. One example of a model pre­diction of nonmonotonic swelling is shown in Fig­ure 17.1 Note that unambiguous interpretations of neutron-irradiation data are often confounded by uncertainties in irradiation temperatures and complex temperature histories.155,156 However, the suppression of swelling by high Nb is clear even in these cases.

Bubble sinks can also play a significant role in the post-incubation swelling rates. Neglecting vacancy emission from large voids, and using the same assump­tions described above, leads to a simple expression for the overall normalized swelling rate S, the rate of

increase in total void volume per unit volume divided by the displacement rate as

S = [v BZdZv/[(Zb + Zv + Zd)(Zd(1 + B) + Zv + Zb)]

[11]

Figure 18 shows S for B = 0.15 and v = 0.3 as a function of Zv/Zd, with a peak at Zd = Zv and

Zb ^ 1, representing the case when nearly all the bubbles have converted to voids and balanced void and dislocation sink strengths. The S decreases at higher and lower Zv/Zd. Figure 18 also shows S as a function of Zv/Zd for a range of Zb/Zv. Increasing Zb with the other sink strengths fixed reduces the S in the limit scaling with 1/Z2. These results again show that significant swelling rates require some

Figure 18 Predicted swelling rate (S) for various bubble to void sink strength ratios (Zb/Zv) as a function of the void to dislocation sink ratio (Zv/Zd). The highest S is for a low Zb/Zv at a balanced void and dislocation sink strengths Zv « Zb. S decreases with increasing Zb/Zv and the corresponding peak rate shifts to lower Zb/Zv.

bubbles to form voids with a sink strength of Zv that is not too small (or large) compared with Zd. However, a large population of unconverted bubbles, with a high sink strength Zb, can greatly reduce swelling rates.

A significant advantage of the CBM is that it requires a relatively modest number of parameters, and parameter combinations, that are generally rea­sonably well known, including for defect production, recombination, dislocation bias, sink strengths, inter­face energy, and Ds& Potential future improvements in modeling bubble and void evolution include better overall parameterization using electronic-atomistic models, a refined equation of state at small bubble sizes, and precipitate specific estimates of Fv based on improved models and direct measurements. Further, it is important to note that the CBM parameters can be estimated experimentally as the pinch-off size between the small bubbles and larger voids.114,124,157

Application of CBM to void swelling requires treatment of the bubble evolution at various sites, including in the matrix, on dislocations, at precipitate interfaces, and in GBs. Increasing the He generation rate (GHe) generally leads to higher bubble concen­trations, scaling as Nb / GHe.m,112,131-133,140,144,172 The exponent p varies between limits of ~0, for totally heterogeneous bubble nucleation on a fixed number of deep trapping sites, to >1 when the dominant He fate is governed by trap binding ener­gies, large He bubble nucleus cluster sizes (most often assumed to be only two atoms), and loss of He to other sinks. Assuming the dominant fate of He is to form matrix bubbles, p has a natural value of ~ 1/2 for the condition that the probability of diffusing He to nucleate a new matrix bubble as a di-He cluster is equal to the probability of the He being absorbed in a previously formed bubble.158

Bubble formation is also sensitive to temperature and depends on the diffusion coefficient and mech­anism, as well as He binding energies at various trapping sites. Substitutional He (Hes) diffuses by vacancy exchange with an activation energy of Ehs ~ 2.4 eV.159 For bimolecular nucleation of matrix bubbles, Nb scales as exp(—Ehs/2kT). Helium can also diffuse as small n > 2 and m > 1 vacancy-He complexes, but bubbles are essentially immobile at much larger sizes. Helium is most likely initially created as interstitial He (Hei), which diffuses so rapidly that it can be considered to simply partition to various trapping sites, including vacancy traps, where Hei + V! Hes. Note that, for interstitial dif­fusion, the matrix concentrations of Hei are so low that migrating Hei-Hei reactions would not be expected to form He bubbles. Thermal detrapping of Hes from vacancies to form Hei is unlikely because of the high thermal binding energy160 and see Section 1.06.5 for other references) but can occur by a Hes + SIA! Hei reaction, as well as by direct displacement events.152,161 If Hei and Hes maintain their identities at trapping sites, they can detrap in the same configura­tion. Clustering reactions between Hes, Hei, and vacancies form bubbles at the trapping sites.

Thus, He binding energies at traps are also critical to the fate of He and the effects of temperature and GHe. Traps include both the microstructural sites noted above as well as deeper local traps within these general sites, such as dislocation jogs and grain boundary junctions136 (and see Section 1.06.5 for other references). If the trapping energies are low, or temperatures are high, He can recycle between various traps and the matrix a number of times before it forms or joins a bubble. However, once formed bubbles are very deep traps, and at a significant sink density, they play a dominant role in the transport and fate of He.

In principle, the binding energies of He clusters are also important to bubble nucleation. Recent ab initio simulations have shown that even small clus­ters of Hei in Fe are bound, although not as strongly as Hes-V complexes. Indeed, the binding energies of small HemVn complexes with n > m are large (2.8­3.8 eV),134’135 suggesting that the bi — or trimolecular bubble nucleation mechanism is a good approximation over a wide range of irradiation conditions. Further, for neutron-irradiation conditions with low GHe and Gdpa that create a vacancy-rich environment, it is also reasonable to assume that He clusters initially evolve along a bubble-dominated path.

As discussed previously, the effects of higher bubble densities on overall microstructural evolutions are complex. The observation that N scales as GHe relation has been used in many parametric studies of the effects of varying bubble and void microstructures. Bubble nucleation and growth and void swelling are sup­pressed at very low GHe. However, as noted above, swelling can sometimes decrease beyond a critical GHe due to higher Nb. Indeed, void formation and swelling can be completely suppressed by a very high concentration of bubbles. High bubble concentrations can also suppress the formation ofdislocation loops and irradiation-enhanced, induced, and modified precipi­tation associated with solute segregation, by keeping

excess concentrations of vacancies and SIA very

low 16,26,111,112,162